Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917052204187648 |
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| author | Chakraborty, Sukrit Goswami, Sayan Patra, Sourav Kanti |
| author_facet | Chakraborty, Sukrit Goswami, Sayan Patra, Sourav Kanti |
| contents | We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability $p\in (0,1)$, we prove that almost surely such a set contains finite sumsets (FS-sets) and finite product sets (FP-sets) of every finite length. In addition, we establish a novel connection between Hindman's partition theorem and the central limit theorem, providing a probabilistic perspective on the asymptotic Gaussian behavior of monochromatic finite sums and products. These results can be interpreted as probabilistic analogues of finite-dimensional versions of Hindman's theorem. Applications, implications, and open questions related to infinite FS-sets and FP-sets are discussed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_01301 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers Chakraborty, Sukrit Goswami, Sayan Patra, Sourav Kanti Combinatorics Probability 05D10, 11B75, 60C05 We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability $p\in (0,1)$, we prove that almost surely such a set contains finite sumsets (FS-sets) and finite product sets (FP-sets) of every finite length. In addition, we establish a novel connection between Hindman's partition theorem and the central limit theorem, providing a probabilistic perspective on the asymptotic Gaussian behavior of monochromatic finite sums and products. These results can be interpreted as probabilistic analogues of finite-dimensional versions of Hindman's theorem. Applications, implications, and open questions related to infinite FS-sets and FP-sets are discussed. |
| title | Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers |
| topic | Combinatorics Probability 05D10, 11B75, 60C05 |
| url | https://arxiv.org/abs/2510.01301 |